math.log10 on complex, real part

Average Error: 32.8 → 17.8

Time: 5.5s

Precision: 64

Math

\[\frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right)}{\log 10}\]

↓

\[\begin{array}{l} \mathbf{if}\;re \le -4.1975508038006968 \cdot 10^{153}:\\ \;\;\;\;\frac{\frac{1}{2}}{\sqrt{\log 10}} \cdot \left(-2 \cdot \left(\log \left(\frac{-1}{re}\right) \cdot \sqrt{\frac{1}{\log 10}}\right)\right)\\ \mathbf{elif}\;re \le -1.53062664724659985 \cdot 10^{-287}:\\ \;\;\;\;\frac{\frac{\frac{1}{2}}{\sqrt{\log 10}}}{\sqrt{\log 10}} \cdot \log \left(re \cdot re + im \cdot im\right)\\ \mathbf{elif}\;re \le 9.88087578424207621 \cdot 10^{-281}:\\ \;\;\;\;\frac{\frac{1}{2}}{\sqrt{\log 10}} \cdot \left(2 \cdot \left(\log im \cdot \sqrt{\frac{1}{\log 10}}\right)\right)\\ \mathbf{elif}\;re \le 2.9332118608681794 \cdot 10^{97}:\\ \;\;\;\;\frac{\frac{\frac{1}{2}}{\sqrt{\log 10}}}{\sqrt{\log 10}} \cdot \log \left(re \cdot re + im \cdot im\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{2}}{\sqrt{\log 10}} \cdot \left(-2 \cdot \left(\log \left(\frac{1}{re}\right) \cdot \sqrt{\frac{1}{\log 10}}\right)\right)\\ \end{array}\]

C

double f(double re, double im) {
        double r96835 = re;
        double r96836 = r96835 * r96835;
        double r96837 = im;
        double r96838 = r96837 * r96837;
        double r96839 = r96836 + r96838;
        double r96840 = sqrt(r96839);
        double r96841 = log(r96840);
        double r96842 = 10.0;
        double r96843 = log(r96842);
        double r96844 = r96841 / r96843;
        return r96844;
}

↓

double f(double re, double im) {
        double r96845 = re;
        double r96846 = -4.197550803800697e+153;
        bool r96847 = r96845 <= r96846;
        double r96848 = 0.5;
        double r96849 = 10.0;
        double r96850 = log(r96849);
        double r96851 = sqrt(r96850);
        double r96852 = r96848 / r96851;
        double r96853 = -2.0;
        double r96854 = -1.0;
        double r96855 = r96854 / r96845;
        double r96856 = log(r96855);
        double r96857 = 1.0;
        double r96858 = r96857 / r96850;
        double r96859 = sqrt(r96858);
        double r96860 = r96856 * r96859;
        double r96861 = r96853 * r96860;
        double r96862 = r96852 * r96861;
        double r96863 = -1.5306266472465998e-287;
        bool r96864 = r96845 <= r96863;
        double r96865 = r96852 / r96851;
        double r96866 = r96845 * r96845;
        double r96867 = im;
        double r96868 = r96867 * r96867;
        double r96869 = r96866 + r96868;
        double r96870 = log(r96869);
        double r96871 = r96865 * r96870;
        double r96872 = 9.880875784242076e-281;
        bool r96873 = r96845 <= r96872;
        double r96874 = 2.0;
        double r96875 = log(r96867);
        double r96876 = r96875 * r96859;
        double r96877 = r96874 * r96876;
        double r96878 = r96852 * r96877;
        double r96879 = 2.9332118608681794e+97;
        bool r96880 = r96845 <= r96879;
        double r96881 = r96857 / r96845;
        double r96882 = log(r96881);
        double r96883 = r96882 * r96859;
        double r96884 = r96853 * r96883;
        double r96885 = r96852 * r96884;
        double r96886 = r96880 ? r96871 : r96885;
        double r96887 = r96873 ? r96878 : r96886;
        double r96888 = r96864 ? r96871 : r96887;
        double r96889 = r96847 ? r96862 : r96888;
        return r96889;
}

Error

Bits error versus re

Bits error versus im

Derivation

1. Split input into 4 regimes

2. if re < -4.197550803800697e+153

   1. Initial program 63.9

      \[\frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right)}{\log 10}\]
   2. Using strategy rm

   3. Applied clear-num 63.9

      \[\leadsto \color{blue}{\frac{1}{\frac{\log 10}{\log \left(\sqrt{re \cdot re + im \cdot im}\right)}}}\]
   4. Using strategy rm

   5. Applied add-cube-cbrt 63.9

      \[\leadsto \frac{\color{blue}{\left(\sqrt[3]{1} \cdot \sqrt[3]{1}\right) \cdot \sqrt[3]{1}}}{\frac{\log 10}{\log \left(\sqrt{re \cdot re + im \cdot im}\right)}}\]

   6. Applied associate-/l* 63.9

      \[\leadsto \color{blue}{\frac{\sqrt[3]{1} \cdot \sqrt[3]{1}}{\frac{\frac{\log 10}{\log \left(\sqrt{re \cdot re + im \cdot im}\right)}}{\sqrt[3]{1}}}}\]

   7. Simplified 63.9

      \[\leadsto \frac{\sqrt[3]{1} \cdot \sqrt[3]{1}}{\color{blue}{\frac{\log 10}{\log \left(\sqrt{re \cdot re + im \cdot im}\right)}}}\]
   8. Using strategy rm

   9. Applied pow1/2 63.9

      \[\leadsto \frac{\sqrt[3]{1} \cdot \sqrt[3]{1}}{\frac{\log 10}{\log \color{blue}{\left({\left(re \cdot re + im \cdot im\right)}^{\frac{1}{2}}\right)}}}\]

   10. Applied log-pow 63.9

       \[\leadsto \frac{\sqrt[3]{1} \cdot \sqrt[3]{1}}{\frac{\log 10}{\color{blue}{\frac{1}{2} \cdot \log \left(re \cdot re + im \cdot im\right)}}}\]

   11. Applied add-sqr-sqrt 63.9

       \[\leadsto \frac{\sqrt[3]{1} \cdot \sqrt[3]{1}}{\frac{\color{blue}{\sqrt{\log 10} \cdot \sqrt{\log 10}}}{\frac{1}{2} \cdot \log \left(re \cdot re + im \cdot im\right)}}\]

   12. Applied times-frac 63.9

       \[\leadsto \frac{\sqrt[3]{1} \cdot \sqrt[3]{1}}{\color{blue}{\frac{\sqrt{\log 10}}{\frac{1}{2}} \cdot \frac{\sqrt{\log 10}}{\log \left(re \cdot re + im \cdot im\right)}}}\]

   13. Applied times-frac 63.9

       \[\leadsto \color{blue}{\frac{\sqrt[3]{1}}{\frac{\sqrt{\log 10}}{\frac{1}{2}}} \cdot \frac{\sqrt[3]{1}}{\frac{\sqrt{\log 10}}{\log \left(re \cdot re + im \cdot im\right)}}}\]

   14. Simplified 63.9

       \[\leadsto \color{blue}{\frac{\frac{1}{2}}{\sqrt{\log 10}}} \cdot \frac{\sqrt[3]{1}}{\frac{\sqrt{\log 10}}{\log \left(re \cdot re + im \cdot im\right)}}\]

   15. Simplified 63.9

       \[\leadsto \frac{\frac{1}{2}}{\sqrt{\log 10}} \cdot \color{blue}{\frac{1}{\frac{\sqrt{\log 10}}{\log \left(re \cdot re + im \cdot im\right)}}}\]

   16. Taylor expanded around -inf 6.8

       \[\leadsto \frac{\frac{1}{2}}{\sqrt{\log 10}} \cdot \color{blue}{\left(-2 \cdot \left(\log \left(\frac{-1}{re}\right) \cdot \sqrt{\frac{1}{\log 10}}\right)\right)}\]

   if -4.197550803800697e+153 < re < -1.5306266472465998e-287 or 9.880875784242076e-281 < re < 2.9332118608681794e+97

   1. Initial program 21.8

      \[\frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right)}{\log 10}\]
   2. Using strategy rm

   3. Applied clear-num 21.8

      \[\leadsto \color{blue}{\frac{1}{\frac{\log 10}{\log \left(\sqrt{re \cdot re + im \cdot im}\right)}}}\]
   4. Using strategy rm

   5. Applied add-cube-cbrt 21.8

      \[\leadsto \frac{\color{blue}{\left(\sqrt[3]{1} \cdot \sqrt[3]{1}\right) \cdot \sqrt[3]{1}}}{\frac{\log 10}{\log \left(\sqrt{re \cdot re + im \cdot im}\right)}}\]

   6. Applied associate-/l* 21.8

      \[\leadsto \color{blue}{\frac{\sqrt[3]{1} \cdot \sqrt[3]{1}}{\frac{\frac{\log 10}{\log \left(\sqrt{re \cdot re + im \cdot im}\right)}}{\sqrt[3]{1}}}}\]

   7. Simplified 21.8

      \[\leadsto \frac{\sqrt[3]{1} \cdot \sqrt[3]{1}}{\color{blue}{\frac{\log 10}{\log \left(\sqrt{re \cdot re + im \cdot im}\right)}}}\]
   8. Using strategy rm

   9. Applied pow1/2 21.8

      \[\leadsto \frac{\sqrt[3]{1} \cdot \sqrt[3]{1}}{\frac{\log 10}{\log \color{blue}{\left({\left(re \cdot re + im \cdot im\right)}^{\frac{1}{2}}\right)}}}\]

   10. Applied log-pow 21.8

       \[\leadsto \frac{\sqrt[3]{1} \cdot \sqrt[3]{1}}{\frac{\log 10}{\color{blue}{\frac{1}{2} \cdot \log \left(re \cdot re + im \cdot im\right)}}}\]

   11. Applied add-sqr-sqrt 21.8

       \[\leadsto \frac{\sqrt[3]{1} \cdot \sqrt[3]{1}}{\frac{\color{blue}{\sqrt{\log 10} \cdot \sqrt{\log 10}}}{\frac{1}{2} \cdot \log \left(re \cdot re + im \cdot im\right)}}\]

   12. Applied times-frac 22.0

       \[\leadsto \frac{\sqrt[3]{1} \cdot \sqrt[3]{1}}{\color{blue}{\frac{\sqrt{\log 10}}{\frac{1}{2}} \cdot \frac{\sqrt{\log 10}}{\log \left(re \cdot re + im \cdot im\right)}}}\]

   13. Applied times-frac 21.8

       \[\leadsto \color{blue}{\frac{\sqrt[3]{1}}{\frac{\sqrt{\log 10}}{\frac{1}{2}}} \cdot \frac{\sqrt[3]{1}}{\frac{\sqrt{\log 10}}{\log \left(re \cdot re + im \cdot im\right)}}}\]

   14. Simplified 21.8

       \[\leadsto \color{blue}{\frac{\frac{1}{2}}{\sqrt{\log 10}}} \cdot \frac{\sqrt[3]{1}}{\frac{\sqrt{\log 10}}{\log \left(re \cdot re + im \cdot im\right)}}\]

   15. Simplified 21.8

       \[\leadsto \frac{\frac{1}{2}}{\sqrt{\log 10}} \cdot \color{blue}{\frac{1}{\frac{\sqrt{\log 10}}{\log \left(re \cdot re + im \cdot im\right)}}}\]
   16. Using strategy rm

   17. Applied associate-/r/ 21.7

       \[\leadsto \frac{\frac{1}{2}}{\sqrt{\log 10}} \cdot \color{blue}{\left(\frac{1}{\sqrt{\log 10}} \cdot \log \left(re \cdot re + im \cdot im\right)\right)}\]

   18. Applied associate-*r* 21.6

       \[\leadsto \color{blue}{\left(\frac{\frac{1}{2}}{\sqrt{\log 10}} \cdot \frac{1}{\sqrt{\log 10}}\right) \cdot \log \left(re \cdot re + im \cdot im\right)}\]

   19. Simplified 21.6

       \[\leadsto \color{blue}{\frac{\frac{\frac{1}{2}}{\sqrt{\log 10}}}{\sqrt{\log 10}}} \cdot \log \left(re \cdot re + im \cdot im\right)\]

   if -1.5306266472465998e-287 < re < 9.880875784242076e-281

   1. Initial program 34.0

      \[\frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right)}{\log 10}\]
   2. Using strategy rm

   3. Applied clear-num 34.0

      \[\leadsto \color{blue}{\frac{1}{\frac{\log 10}{\log \left(\sqrt{re \cdot re + im \cdot im}\right)}}}\]
   4. Using strategy rm

   5. Applied add-cube-cbrt 34.0

      \[\leadsto \frac{\color{blue}{\left(\sqrt[3]{1} \cdot \sqrt[3]{1}\right) \cdot \sqrt[3]{1}}}{\frac{\log 10}{\log \left(\sqrt{re \cdot re + im \cdot im}\right)}}\]

   6. Applied associate-/l* 34.0

      \[\leadsto \color{blue}{\frac{\sqrt[3]{1} \cdot \sqrt[3]{1}}{\frac{\frac{\log 10}{\log \left(\sqrt{re \cdot re + im \cdot im}\right)}}{\sqrt[3]{1}}}}\]

   7. Simplified 34.0

      \[\leadsto \frac{\sqrt[3]{1} \cdot \sqrt[3]{1}}{\color{blue}{\frac{\log 10}{\log \left(\sqrt{re \cdot re + im \cdot im}\right)}}}\]
   8. Using strategy rm

   9. Applied pow1/2 34.0

      \[\leadsto \frac{\sqrt[3]{1} \cdot \sqrt[3]{1}}{\frac{\log 10}{\log \color{blue}{\left({\left(re \cdot re + im \cdot im\right)}^{\frac{1}{2}}\right)}}}\]

   10. Applied log-pow 34.0

       \[\leadsto \frac{\sqrt[3]{1} \cdot \sqrt[3]{1}}{\frac{\log 10}{\color{blue}{\frac{1}{2} \cdot \log \left(re \cdot re + im \cdot im\right)}}}\]

   11. Applied add-sqr-sqrt 34.0

       \[\leadsto \frac{\sqrt[3]{1} \cdot \sqrt[3]{1}}{\frac{\color{blue}{\sqrt{\log 10} \cdot \sqrt{\log 10}}}{\frac{1}{2} \cdot \log \left(re \cdot re + im \cdot im\right)}}\]

   12. Applied times-frac 34.1

       \[\leadsto \frac{\sqrt[3]{1} \cdot \sqrt[3]{1}}{\color{blue}{\frac{\sqrt{\log 10}}{\frac{1}{2}} \cdot \frac{\sqrt{\log 10}}{\log \left(re \cdot re + im \cdot im\right)}}}\]

   13. Applied times-frac 33.9

       \[\leadsto \color{blue}{\frac{\sqrt[3]{1}}{\frac{\sqrt{\log 10}}{\frac{1}{2}}} \cdot \frac{\sqrt[3]{1}}{\frac{\sqrt{\log 10}}{\log \left(re \cdot re + im \cdot im\right)}}}\]

   14. Simplified 33.9

       \[\leadsto \color{blue}{\frac{\frac{1}{2}}{\sqrt{\log 10}}} \cdot \frac{\sqrt[3]{1}}{\frac{\sqrt{\log 10}}{\log \left(re \cdot re + im \cdot im\right)}}\]

   15. Simplified 33.9

       \[\leadsto \frac{\frac{1}{2}}{\sqrt{\log 10}} \cdot \color{blue}{\frac{1}{\frac{\sqrt{\log 10}}{\log \left(re \cdot re + im \cdot im\right)}}}\]

   16. Taylor expanded around 0 32.3

       \[\leadsto \frac{\frac{1}{2}}{\sqrt{\log 10}} \cdot \color{blue}{\left(2 \cdot \left(\log im \cdot \sqrt{\frac{1}{\log 10}}\right)\right)}\]

   if 2.9332118608681794e+97 < re

   1. Initial program 51.5

      \[\frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right)}{\log 10}\]
   2. Using strategy rm

   3. Applied clear-num 51.5

      \[\leadsto \color{blue}{\frac{1}{\frac{\log 10}{\log \left(\sqrt{re \cdot re + im \cdot im}\right)}}}\]
   4. Using strategy rm

   5. Applied add-cube-cbrt 51.5

      \[\leadsto \frac{\color{blue}{\left(\sqrt[3]{1} \cdot \sqrt[3]{1}\right) \cdot \sqrt[3]{1}}}{\frac{\log 10}{\log \left(\sqrt{re \cdot re + im \cdot im}\right)}}\]

   6. Applied associate-/l* 51.5

      \[\leadsto \color{blue}{\frac{\sqrt[3]{1} \cdot \sqrt[3]{1}}{\frac{\frac{\log 10}{\log \left(\sqrt{re \cdot re + im \cdot im}\right)}}{\sqrt[3]{1}}}}\]

   7. Simplified 51.5

      \[\leadsto \frac{\sqrt[3]{1} \cdot \sqrt[3]{1}}{\color{blue}{\frac{\log 10}{\log \left(\sqrt{re \cdot re + im \cdot im}\right)}}}\]
   8. Using strategy rm

   9. Applied pow1/2 51.5

      \[\leadsto \frac{\sqrt[3]{1} \cdot \sqrt[3]{1}}{\frac{\log 10}{\log \color{blue}{\left({\left(re \cdot re + im \cdot im\right)}^{\frac{1}{2}}\right)}}}\]

   10. Applied log-pow 51.5

       \[\leadsto \frac{\sqrt[3]{1} \cdot \sqrt[3]{1}}{\frac{\log 10}{\color{blue}{\frac{1}{2} \cdot \log \left(re \cdot re + im \cdot im\right)}}}\]

   11. Applied add-sqr-sqrt 51.5

       \[\leadsto \frac{\sqrt[3]{1} \cdot \sqrt[3]{1}}{\frac{\color{blue}{\sqrt{\log 10} \cdot \sqrt{\log 10}}}{\frac{1}{2} \cdot \log \left(re \cdot re + im \cdot im\right)}}\]

   12. Applied times-frac 51.5

       \[\leadsto \frac{\sqrt[3]{1} \cdot \sqrt[3]{1}}{\color{blue}{\frac{\sqrt{\log 10}}{\frac{1}{2}} \cdot \frac{\sqrt{\log 10}}{\log \left(re \cdot re + im \cdot im\right)}}}\]

   13. Applied times-frac 51.5

       \[\leadsto \color{blue}{\frac{\sqrt[3]{1}}{\frac{\sqrt{\log 10}}{\frac{1}{2}}} \cdot \frac{\sqrt[3]{1}}{\frac{\sqrt{\log 10}}{\log \left(re \cdot re + im \cdot im\right)}}}\]

   14. Simplified 51.5

       \[\leadsto \color{blue}{\frac{\frac{1}{2}}{\sqrt{\log 10}}} \cdot \frac{\sqrt[3]{1}}{\frac{\sqrt{\log 10}}{\log \left(re \cdot re + im \cdot im\right)}}\]

   15. Simplified 51.5

       \[\leadsto \frac{\frac{1}{2}}{\sqrt{\log 10}} \cdot \color{blue}{\frac{1}{\frac{\sqrt{\log 10}}{\log \left(re \cdot re + im \cdot im\right)}}}\]

   16. Taylor expanded around inf 8.0

       \[\leadsto \frac{\frac{1}{2}}{\sqrt{\log 10}} \cdot \color{blue}{\left(-2 \cdot \left(\log \left(\frac{1}{re}\right) \cdot \sqrt{\frac{1}{\log 10}}\right)\right)}\]
3. Recombined 4 regimes into one program.

4. Final simplification 17.8

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \le -4.1975508038006968 \cdot 10^{153}:\\ \;\;\;\;\frac{\frac{1}{2}}{\sqrt{\log 10}} \cdot \left(-2 \cdot \left(\log \left(\frac{-1}{re}\right) \cdot \sqrt{\frac{1}{\log 10}}\right)\right)\\ \mathbf{elif}\;re \le -1.53062664724659985 \cdot 10^{-287}:\\ \;\;\;\;\frac{\frac{\frac{1}{2}}{\sqrt{\log 10}}}{\sqrt{\log 10}} \cdot \log \left(re \cdot re + im \cdot im\right)\\ \mathbf{elif}\;re \le 9.88087578424207621 \cdot 10^{-281}:\\ \;\;\;\;\frac{\frac{1}{2}}{\sqrt{\log 10}} \cdot \left(2 \cdot \left(\log im \cdot \sqrt{\frac{1}{\log 10}}\right)\right)\\ \mathbf{elif}\;re \le 2.9332118608681794 \cdot 10^{97}:\\ \;\;\;\;\frac{\frac{\frac{1}{2}}{\sqrt{\log 10}}}{\sqrt{\log 10}} \cdot \log \left(re \cdot re + im \cdot im\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{2}}{\sqrt{\log 10}} \cdot \left(-2 \cdot \left(\log \left(\frac{1}{re}\right) \cdot \sqrt{\frac{1}{\log 10}}\right)\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020046 
(FPCore (re im)
  :name "math.log10 on complex, real part"
  :precision binary64
  (/ (log (sqrt (+ (* re re) (* im im)))) (log 10)))